Topics+for+Semester+1+Exam

__**Topics for Semester 1 Exam - January 2012**__

SEQUENCES AND SERIES: arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series; sigma notation

COMBINATORICS: counting principles, including permutations and combinations; binomial theorem: expansion of (//a// + //b//)// n //

FUNCTIONS: Concept of function //f//(//x//) : domain, range; image (value); composite functions //f o g// ; identity function; inverse function //f// **−**1; t he graph of a function; its equation //y// = //f//(//x//); function graphing skills: use of a GDC to graph a variety of functions; investigation of key features of graphs; solutions of equations graphically; transformations of graphs: translations; stretches; reflections in the axes; the graph of //y// = //f// −1 (//x//) as the reflection in the line //y// = //x// of the graph of //y// = //f//(//x//); the graph of 1 / //f//(//x//) from //y// = //f//(//x//); the graphs of the absolute value functions, y =| //f//(//x//) | and y = //f//( |//x|// ); the reciprocal function //f//(//x//) = 1 / //x//, //x//≠ 0: its graph; its self-inverse nature; inequalities in one variable, using their graphical representation; the factor and remainder theorems, with application to the solution of polynomial equations and inequalities.

QUADRATICS: the quadratic function //f(//x//) = ax// 2 + //b//x //+// c : its graph; axis of symmetry //x// = –//b// / 2//a// ; the form //a//(//x// – //h//) 2 + //k//; the form //a//(//x// – //p//)(//x// – //q//); the solution of //ax// 2 + //b//x //+// c = 0, //a// ≠ 0; the quadratic formula; use of the discriminant ∆ = //b// 2 – 4//ac//

CIRCLES AND TRIGONOMETRY: the circle: radian measure of angles; length of an arc; area of a sector; Solution of triangles: cosine rule, sine rule, area of triangle as 1/2 //ab sin C//

CALCULUS: informal ideas of limit and convergence; differentiation by first principles, derivative of //x n //, derivative interpreted as a gradient function and as a rate of change; differentiation of a sum and a real multiple of functions; the second and higher derivatives; local maximum and minimum points; use of the first and second derivatives in optimization (max/min) problems; indefinite integration as anti-differentiation; indefinite integral of //x n //; anti-differentiation with a boundary condition to determine the constant term; definite intergrals; area between a curve and the //x//-axis and //y//-axis in a given interval; areas between curves; kinematic problems involving displacement, velocity and acceleration; the significance of the second derivative; distinction between maximum and minimum points; points of inflexion with zero and non-zero gradients